Solving the Number Analogy 12:156 :: 14:?

This question asks us to find a relationship between the first pair of numbers (12 and 156) and apply the same relationship to the first number of the second pair (14) to find the missing second number.

Analyzing the Relationship in the First Pair: 12 and 156

Let's look for a pattern connecting 12 and 156. Often, in such analogies, the relationship involves basic arithmetic operations like addition, subtraction, multiplication, division, squares, cubes, or operations involving the number itself or its neighbours.

Let's try some common patterns:

• Is 156 a multiple of 12? $156 \div 12$. $12 \times 10 = 120$. $156 - 120 = 36$. $12 \times 3 = 36$. So, $12 \times 13 = 156$.
• How is 13 related to 12? 13 is one more than 12 ($12+1$).

So, the relationship seems to be: The second number is the first number multiplied by (the first number + 1).

Using mathematical notation, if the first number is $n$, the second number is $n \times (n+1)$.

Let's verify this with the first pair, where $n=12$:

Second number $= 12 \times (12+1) = 12 \times 13$

Calculation: $12 \times 13 = 156$. This matches the given second number in the first pair.

Applying the Pattern to the Second Pair: 14 and ?

Now we apply the same rule to the second pair, where the first number is 14. Let the missing number be $X$.

Here, $n=14$. According to the pattern, the missing number $X$ should be $n \times (n+1)$.

Missing number $X = 14 \times (14+1)$

Missing number $X = 14 \times 15$

Let's calculate $14 \times 15$:

$14 \times 15 = 14 \times (10 + 5) = (14 \times 10) + (14 \times 5)$

$14 \times 10 = 140$

$14 \times 5 = 70$

$140 + 70 = 210$

So, the missing number is 210.

Comparing with the Options

Let's check the given options:

• Option 1: 195
• Option 2: 205
• Option 3: 208
• Option 4: 210

Our calculated missing number, 210, matches Option 4.

Conclusion

The relationship in the analogy 12 : 156 :: 14 : ? is that the second number is obtained by multiplying the first number by one more than itself. Applying this rule to 14 gives $14 \times (14+1) = 14 \times 15 = 210$.

Therefore, the related number is 210.

Revision Table: Number Analogy

Additional Information: Solving Numerical Reasoning Problems

Numerical reasoning questions, like number analogies, test your ability to identify patterns and logical relationships between numbers. Here are some common strategies:

• Look for arithmetic relationships: Addition, subtraction, multiplication, division between the numbers.
• Consider squares and cubes: The numbers might be squares, cubes, or related to squares/cubes (e.g., $n^2 \pm a$, $n^3 \pm a$). In this problem, 156 is close to $12^2 = 144$, but $156 = 144 + 12$. This is another way to see the pattern: $n^2 + n = n(n+1)$.
• Check for sequences: If there are multiple pairs, look for patterns across the pairs.
• Think about digits: Sometimes the pattern involves the digits of the number itself (sum of digits, product of digits, etc.).
• Try simple relationships first: Start with the most straightforward possibilities before moving to more complex ones.

Practice is key to becoming familiar with different types of number patterns.